The present work develops a novel and rigorous theory of vorticity decomposition within fluid kinematics. At a point where the velocity gradient tensor (VGT) A3D has a positive discriminant, the local rotation axis e3 is uniquely defined as the real eigenvector corresponding to the real eigenvalue of A3D, with the sign convention ω3≡ω·e30 for the axial vorticity component. On the rotation-axis-normal plane P, a pair of Lagrangian functions (L,G) is constructed, whose objective functions are the total and specific angular velocities, (WL(e),WD(e)), of the unit orientation vector e for an in-plane material line element. These angular velocities depend solely on the restricted 2D VGT A≡A2D on P, whose extremum problems are shown to be equivalent to the eigenvalue problems for a pair of commutative angular-velocity operators (Ψ,Γ). We find that Γ resembles the spin angular-momentum operator for an electron in quantum mechanics, with its eigenvectors analogous to two-component spinors. Correspondingly, we introduce a pair of commutative vorticity operators (ΨR,ΓS)≡(2Ψ,−2Γ) and propose an operator-form vorticity decomposition (OVD) ω3σ0=ΨR+ΓS, where σ0≡I2D is the identity tensor; ΨR and ΓS are respectively termed the rigid-rotation and spin vorticity operators. Importantly, the eigenvalues of (ΨR,ΓS) are precisely the characteristic vorticity modes (2ψ±,γ±) in the existing invariant vorticity decompositions (IVDs), while their common eigenvectors just constitute the orthonormal NND (normal-nilpotent decomposition) bases used for obtaining the real Schur forms of the VGT. This novel result serves as the unifying principle underlying previously distinct IVDs, which underscores the essential roles of both the IVD vorticity modes and the NND bases in formulating a complete theory of kinematic vorticity decomposition. Physically, the theory indicates a potential for the coexistence of both KKB (Klein–Kaden–Betz) and anti-KKB mechanisms in a swirling flow system under certain conditions. Moreover, the applicability of the proposed vorticity-operator pair is extended to cases of non-positive discriminant and zero vorticity. These findings could offer valuable insights into the physics underlying diverse vortical-flow phenomena.
Tao Chen (Sun,) studied this question.